A Short History of Probability Dr. Alan M. Polansky Division of Statistics Northern Illinois University French Society in the 1650’s ! Gambling was popular and fashionable ! Not restricted by law ! As the games became more complicated and the stakes became larger there was a need for mathematical methods for computing chances. History of Probability 2 Enter the Mathematicians ! A well-known gambler, the chevalier De Mere consulted Blaise Pascal in Paris about a some questions about some games of chance. ! Pascal began to correspond with his friend Pierre Fermat about these problems. History of Probability 3 Classical Probability ! The correspondence between Pascal and Fermat is the origin of the mathematical study of probability. ! The method they developed is now called the classical approach to computing probabilities. ! The method: Suppose a game has n equally likely outcomes, of which m outcomes correspond to winning. Then the probability of winning is m/n. History of Probability 4 Problems with the Classical Method ! The classical method requires a game to be broken down into equally likely outcomes. ! It is not always possible to do this. ! It is not always clear when possibilities are equally likely. History of Probability 5 Experience ! Another method, known as the frequency method had also been used for some time. ! This method consists of repeating a game a large number of times under the same conditions. The probability of winning is then approximately equal to the proportion of wins in the repeats. ! This method was used by Pascal and Fermat to verify results obtained by the classical method. History of Probability 6 Early Generalizations ! James Bernoulli proved that the frequency method and the classical method are consistent with one another in his book Ars Conjectandi in 1713. History of Probability 7 Early Generalizations ! Abraham De Moivre provided many tools to make the classical method more useful, including the multiplication rule, in his book The Doctrine of Chances in 1718. ! The book was popular, eventually going through three editions. History of Probability 8 From Games to Science th ! Throughout the 18 century, the application of probability moved from games of chance to scientific problems: ! Mathematical theory of life insurance - life tables. ! Biological problems - what is the probability of being born female or male? History of Probability 9 Applied Probability ! Pierre-Simon Laplace presented a mathematical theory of probability with an emphasis on scientific applications in his 1812 book Theorie Analytique des Probabilities. ! Unfortunately, Laplace only considered the classical method, leaving no indication on how the method was to be applied to general problems.History of Probability 10 Stagnation the Frustration ! After the publication of Laplace’s book, the mathematical development of probability stagnated for many years. ! By 1850, many mathematicians found the classical method to be unrealistic for general use and were attempting to redefine probability in terms of the frequency method. ! These attempts were never fully accepted and the stagnation continued. History of Probability 11 Axiomatic Development ! Andrey Kolmogorov developed the first rigorous approach to probability in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnun. ! He built up probability theory from fundamental axioms in a way comparable with Euclid's treatment of geometry. History of Probability 12 Probability Today ! Modern research in probability theory is closely related to the mathematical field of measure theory. ! Modern innovators in the field include Patrick Billingsley (University of Chicago), Yuan Shih Chow (Columbia), Kai Lai Chung (Stanford), Samuel Karlin (Stanford), Rolf-Dieter Reiss, Sheldon Ross (Berkeley), Henry Teicher (Rutgers) and many many more… History of Probability 13.